Charge-orbital ordering in low-temperature structures of magnetite:GGA+Uinvestigations

The atomic and electronic structure of magnetite $({\mathrm{Fe}}_{3}{\mathrm{O}}_{4})$ in the four possible low-temperature structures, namely, $P2∕c\ensuremath{-}Pmca$ (I), $Pmca$ (II), $Pmc{2}_{1}$ (III), and $Cc$ (IV), have been investigated by generalized gradient $\text{approximation}+\text{Hubbard}$ $U$ $(\mathrm{GGA}+U)$ electronic structure and structural optimization calculations. Charge-orbital ordering is found to exist in all the four structures. The charge-orbital ordering and hence the Verwey metal-insulator transition is shown to be driven by the on-site Fe $d$-electron correlation. The theoretical charge-orbital ordering patterns in the I, II, and III structures do not satisfy the Anderson criterion but are consistent with recent neutron and x-ray diffraction experiments. The IV $(Cc)$ structure is found to be the ground state structure. In the IV structure, the charge-orbital ordering on $3∕4$ of the tetrahedra does not satisfy the Anderson condition, while on $1∕4$ of the tetrahedra it does. The observed entropy change at the Verwey transition, which has been a long standing puzzle, is analyzed and found to be consistent with the charge-orbital orders obtained here.